A new method for digital true random number generation based on asynchronous logic circuits with feedback is introduced. In particular, a concrete technique using the so-called Galois and Fibonacci ring oscillators is developed and analyzed both theoretically and experimentally. The generated random binary sequences may have a very high speed and a higher and more robust entropy rate in comparison with previous proposals for digital random number generators. A new method for digital postprocessing of random data based on irregularly clocked nonautonomous synchronous logic circuits with feedback is also introduced and a concrete technique using a self-clock-controlled linear feedback shift register is proposed. The postprocessing can provide both randomness extraction and computationally secure speed increase of input random data.